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Math Help - Coordinate Geometry

  1. #1
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    Coordinate Geometry



    In the diagram above, triangle ACD is such that the coordinates of A and B are (2, 12) and (6, 9) respectively. BE is parallel to CD and BC = \frac{2}{3}AC . The gradient of BE is \frac{2}{7}. The equation of AD is 3y=5x+26.

    Calculate the coordinates of C.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let M_1(x_1,y_1), \ M_2(x_2,y_2) be two points and M(x_M,y_M) a point on the segment M_1M_2 such as \displaystyle\frac{M_1M}{MM_2}=k.

    Then x_M=\displaystyle\frac{x_1+kx_2}{1+k}, \ y_M=\displaystyle\frac{y_1+ky_2}{1+k}

    In this case \displaystyle\frac{AB}{AC}=\frac{1}{2}

    Then x_B=\displaystyle\frac{x_A+\frac{1}{2}x_C}{1+\frac  {1}{2}}

    y_B=\displaystyle\frac{y_A+\frac{1}{2}y_C}{1+\frac  {1}{2}}

    You know the coordinates of A and B, so you can find the coordinates of C.
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  3. #3
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    Hello, Punch!

    I assume there are more parts to this problem.
    There is way too much information . . .


    In the diagram, \Delta ACD has points: A(2,12) and B(6,9).

    BE is parallel to CD and BC = \frac{2}{3}AC .
    The gradient of BE is \frac{2}{7}.
    The equation of AD is: 3y\:=\:5x+26.

    Calculate the coordinates of C.

    Here is a simplified diagram:


    Code:
          |
          |       A
          | (2,12)o - - +
          |        \    :
          |         \   :
          |          \  :
          |           \ :
          |            \:
          |       B(6,9)o - - - - +
          |              \        :
          |               \       :
          |                \      :
          |                 \     :
          |                  \    :
          |                   \   :
          |                    \  :
          |                     \ :
          |                      \:
          |                     C o
          |
      - - + - - - - - - - - - - - - - -
          |

    \text{Since }BC = \frac{2}{3}AC,\; BC\text{ is twice }AB

    Going from A to B, we move: 4 units right and 3 units down.

    So, going from B to C, we move: 8 units right and 6 units down.


    Therefore, point C is (14,3).

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  4. #4
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    you are right soroban
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